When it comes to animation principles and body mechanics "rules," there are just far too many to list. We spend 18 months teaching these rules and methods at Animation Mentor, so I just can't possibly fit that stuff into a blog post, but I will say that the 12 principles of animation as laid out by Frank Thomas and Ollie Johnston in their seminal book, The Illusion of Life, is a good place to start.
When I was a newer animation student, in fact, I had a list of those principles taped to my animation disc and also to my computer monitor. As I would animate, I would look at it occasionally, and make sure that each of those principles was being properly addressed in my work.
I had that on my monitor at ILM as recently as a few years ago, in fact, and it really wouldn't be a bad idea for me to have it up there still! It comes in handy, just as a good reminder, especially as I'm planning or blocking my work in. Just being able to glance up and see the words "line of action" can help me notice that I'm missing something in my work.
One thing I've learned through the process of helping create Animation Mentor is just how deep the "basic fundamental principles" of animation are, and how deep they can go. Interviewing so many amazing animators for the video lectures at the school was a shocking experience for me, because I was continually learning new things about principles that I thought I already knew! It was a really cool by-product of being a part of the school, and a good reminder that we will NEVER know everything there is to know about animation. It's simply too deep and complex...
But man, that just makes it even cooler! It'll never get boring, and there will always be new discoveries waiting for us!